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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
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(load all 4 compositions to match this statistic)
Values
[1,2,3] => ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000264
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 25%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 3 - 2
[1,3,2] => [1,3,2] => [1,2,3] => ([],3)
=> ? = 2 - 2
[2,1,3] => [2,1,3] => [1,2,3] => ([],3)
=> ? = 2 - 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 3 - 2
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? = 2 - 2
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? = 2 - 2
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 2
[1,4,2,3] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 2
[1,4,3,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 2
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? = 2 - 2
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([],4)
=> ? = 3 - 2
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 2
[3,1,2,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 2
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 3 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 - 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 - 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 - 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 - 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,5,2,3,4] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,5,2,4,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 2
[2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 - 2
[2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 - 2
[2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[2,4,1,3,5] => [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 - 2
[2,4,1,5,3] => [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 2
[2,4,3,1,5] => [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,3,5,6,4,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,3,6,4,5,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,4,5,6,2,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,4,5,6,3,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,4,6,2,5,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,4,6,3,5,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,4,6,5,2,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,4,6,5,3,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,3,6,2,4] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,3,6,4,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,4,6,2,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,4,6,3,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,6,2,3,4] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,6,2,4,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,6,3,2,4] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,6,3,4,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,6,4,2,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,5,6,4,3,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,3,4,2,5] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,3,4,5,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,3,5,2,4] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,3,5,4,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,4,2,3,5] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,4,2,5,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,4,3,2,5] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,4,3,5,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,4,5,2,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,4,5,3,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,5,2,3,4] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,5,2,4,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,5,3,2,4] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,5,3,4,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,5,4,2,3] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[1,6,5,4,3,2] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[2,4,5,3,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[2,5,3,4,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[2,5,4,3,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,1,4,2,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,4,1,2,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,4,2,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,2,5,1,3,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,2,5,3,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,5,1,2,3,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,5,1,3,2,6] => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00325: Permutations —ones to leading⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[1,3,2] => [2,3,1] => [1,2,3] => ([],3)
=> ? = 2 + 1
[2,1,3] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 1
[1,2,4,3] => [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? = 2 + 1
[1,3,2,4] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,3,4,2] => [2,3,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 + 1
[1,4,2,3] => [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 1
[1,4,3,2] => [3,4,2,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
[2,1,3,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,4,3] => [2,4,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 + 1
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 1
[3,1,2,4] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 1
[3,2,1,4] => [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? = 2 + 1
[1,2,4,3,5] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,4,5,3] => [2,3,4,1,5] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [3,4,5,1,2] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,4,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 + 1
[1,3,4,2,5] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,4,5,2] => [2,3,1,5,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,3,5,2,4] => [3,4,1,2,5] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,3,5,4,2] => [3,4,2,1,5] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,4,2,3,5] => [1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [2,3,5,1,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,3,2,5] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,3,5,2] => [2,3,1,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 4 + 1
[1,4,5,2,3] => [3,4,1,5,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,4,5,3,2] => [3,4,2,5,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 4 + 1
[1,5,2,3,4] => [4,5,1,3,2] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,5,2,4,3] => [4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[1,5,3,2,4] => [4,5,2,3,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[1,5,3,4,2] => [4,5,3,2,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,5,4,2,3] => [4,5,2,1,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,5,4,3,2] => [4,5,3,1,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[2,1,3,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,3,5,4] => [2,4,3,5,1] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 + 1
[2,1,4,3,5] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,4,5,3] => [2,4,3,1,5] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [3,5,4,1,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,4,3] => [3,5,4,2,1] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,4,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => [2,5,3,4,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[2,4,1,3,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,4,1,5,3] => [2,5,3,1,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[3,1,2,4,5] => [1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,2,5,4] => [2,4,5,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,4,2,5] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,4,1,2,5] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,4,2,1,5] => [1,5,4,2,3] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,2,3,1,5] => [1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,3,1,2,5] => [1,4,5,3,2] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,3,2,1,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,6,2,3,4,5] => [5,6,1,4,2,3] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,6,2,3,5,4] => [5,6,1,3,2,4] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,6,2,4,3,5] => [5,6,1,4,3,2] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,6,2,4,5,3] => [5,6,1,3,4,2] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,6,2,5,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[1,6,2,5,4,3] => [5,6,1,2,4,3] => [4,6,5,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,4,5,1,2,6] => [1,5,6,2,3,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,4,5,2,1,6] => [1,6,5,2,3,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,5,1,4,2,6] => [1,4,6,2,5,3] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,5,2,4,1,6] => [1,6,4,2,5,3] => [4,6,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,5,4,1,2,6] => [1,5,6,2,4,3] => [4,6,5,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,5,4,2,1,6] => [1,6,5,2,4,3] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,2,5,1,3,6] => [1,5,3,6,2,4] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,2,5,3,1,6] => [1,6,3,5,2,4] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,3,5,1,2,6] => [1,5,6,3,2,4] => [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,3,5,2,1,6] => [1,6,5,3,2,4] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,5,1,2,3,6] => [1,4,5,6,2,3] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,5,1,3,2,6] => [1,4,6,5,2,3] => [5,6,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,5,2,1,3,6] => [1,5,4,6,2,3] => [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,5,2,3,1,6] => [1,6,4,5,2,3] => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,5,3,1,2,6] => [1,5,6,4,2,3] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[4,5,3,2,1,6] => [1,6,5,4,2,3] => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,2,3,4,1,6] => [1,6,3,4,5,2] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,2,4,1,3,6] => [1,5,3,6,4,2] => [6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,2,4,3,1,6] => [1,6,3,5,4,2] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,3,1,4,2,6] => [1,4,6,3,5,2] => [6,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,3,2,4,1,6] => [1,6,4,3,5,2] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,3,4,1,2,6] => [1,5,6,3,4,2] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,3,4,2,1,6] => [1,6,5,3,4,2] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,1,2,3,6] => [1,4,5,6,3,2] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,1,3,2,6] => [1,4,6,5,3,2] => [6,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,2,1,3,6] => [1,5,4,6,3,2] => [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,2,3,1,6] => [1,6,4,5,3,2] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,3,1,2,6] => [1,5,6,4,3,2] => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,4,3,2,1,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000741
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 3 - 1
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3 - 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,5,3,2,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,5,4,2,3] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 - 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 - 1
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 2 - 1
[2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 - 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,4,3,1,5] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,1,2,4,5] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 - 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 2 - 1
[3,2,4,1,5] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,4,2,1,5] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,1,2,3,5] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[4,1,3,2,5] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[4,2,1,3,5] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,1,2,5] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4,3,2,1,5] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,4,3,5,6] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3 - 1
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,4,6,5,3] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3 - 1
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 2 - 1
[1,2,5,4,6,3] => [3,6,4,5,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3 - 1
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3 - 1
[1,2,5,6,4,3] => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 - 1
[1,5,6,4,3,2] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,6,4,5,3,2] => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,6,5,3,4,2] => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,6,5,4,2,3] => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,6,5,4,3,2] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,5,3,2,1,6] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,3,4,2,1,6] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,2,3,1,6] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,1,2,6] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5,4,3,2,1,6] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2
[3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,1,3,2,5] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
=> [[.,[[[.,[.,.]],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,1,2,4,3,6] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,1,4,2,3,6] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,1,4,3,2,6] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,4,1,2,3,6] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,4,1,3,2,6] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,4,3,1,2,6] => [[[[.,[.,.]],.],.],[.,.]]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3 + 1
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,3] => [[.,.],[.,.]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4 + 1
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 2 + 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 2 + 1
[3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,1,3,2,5] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
[4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
=> [[.,[[[.,[.,.]],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,1,2,4,3,6] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,1,4,2,3,6] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,1,4,3,2,6] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,4,1,2,3,6] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,4,1,3,2,6] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,4,3,1,2,6] => [[[[.,[.,.]],.],.],[.,.]]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000771
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
St000771: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
[1,2,3] => ([],3)
=> ? = 3 - 1
[1,3,2] => ([(1,2)],3)
=> ? = 2 - 1
[2,1,3] => ([(1,2)],3)
=> ? = 2 - 1
[1,2,3,4] => ([],4)
=> ? = 3 - 1
[1,2,4,3] => ([(2,3)],4)
=> ? = 2 - 1
[1,3,2,4] => ([(2,3)],4)
=> ? = 2 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 3 - 1
[2,1,3,4] => ([(2,3)],4)
=> ? = 2 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 3 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 3 - 1
[1,2,3,4,5] => ([],5)
=> ? = 3 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ? = 2 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
[1,2,3] => ([],3)
=> ? = 3 - 1
[1,3,2] => ([(1,2)],3)
=> ? = 2 - 1
[2,1,3] => ([(1,2)],3)
=> ? = 2 - 1
[1,2,3,4] => ([],4)
=> ? = 3 - 1
[1,2,4,3] => ([(2,3)],4)
=> ? = 2 - 1
[1,3,2,4] => ([(2,3)],4)
=> ? = 2 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 3 - 1
[2,1,3,4] => ([(2,3)],4)
=> ? = 2 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 3 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 3 - 1
[1,2,3,4,5] => ([],5)
=> ? = 3 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ? = 2 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 2 - 1
[3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000259
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
[1,2,3] => ([],3)
=> ([],3)
=> ? = 3
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4] => ([],4)
=> ([],4)
=> ? = 3
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? = 3
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Values
[1,2,3] => ([],3)
=> ([],3)
=> ? = 3
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2
[1,2,3,4] => ([],4)
=> ([],4)
=> ? = 3
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? = 3
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
[4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
[4,1,3,6,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000302The determinant of the distance matrix of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
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